$$ \begin {aligned} \int _{0}^{\infty }\frac {\cos tx}{\cosh \frac {\pi x}2}dx &=\frac {2}\pi \int _{0}^{\infty }\frac {\cos \frac {2tx}\pi }{\cosh x}dx\\ &=\frac {1}\pi \int _{-\infty }^{\infty }\frac {\cos \frac {2tx}\pi }{\cosh x}dx\\ &=\frac {1}\pi \int _{-\infty }^{\infty }\frac {e^{\frac {2txi}\pi }+e^{-\frac {2txi}\pi }}{1+e^{2x}}e^{x}dx\\ &=\frac {1}\pi \int _{0}^{\infty }\frac {u^{\frac {2ti}\pi }+u^{-\frac {2ti}\pi }}{1+u^2}du\\ &=\frac {1}{2\pi }\int _{0}^{\infty }\frac {s^{\frac {ti}\pi -\frac {1}2}+s^{-\frac {ti}\pi -\frac {1}2}}{1+s}ds\\ &=\frac {1}{2\pi }\left (\frac {\pi }{\sin \pi \left (\frac {ti}{2\pi }+\frac {1}2\right )}+\frac {\pi }{\sin \pi \left (-\frac {ti}{2\pi }+\frac {1}{2}\right )}\right )\\ &=\frac {1}2\left (\frac {1}{\cosh t}+\frac {1}{\cosh \left (-t\right )}\right )\\ &=\frac {1}{\cosh t} \end {aligned} $$